<h2>Problem 93</h2>
<div style="color:#666;font-size:80%;">15 April 2005</div><br />
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<p>By using each of the digits from the set, {1, 2, 3, 4}, exactly once, and making use of the four arithmetic operations (+, <img src='images/symbol_minus.gif' width='9' height='3' alt='&minus;' border='0' style='vertical-align:middle;' />, *, /) and brackets/parentheses, it is possible to form different positive integer targets.</p>
<p>For example,</p>
<p style='margin-left:50px;font-family:courier new;'>8 = (4 * (1 + 3)) / 2<br />
14 = 4 * (3 + 1 / 2)<br />
19 = 4 * (2 + 3) <img src='images/symbol_minus.gif' width='9' height='3' alt='&minus;' border='0' style='vertical-align:middle;' /> 1<br />
36 = 3 * 4 * (2 + 1)</p>
<p>Note that concatenations of the digits, like 12 + 34, are not allowed.</p>
<p>Using the set, {1, 2, 3, 4}, it is possible to obtain thirty-one different target numbers of which 36 is the maximum, and each of the numbers 1 to 28 can be obtained before encountering the first non-expressible number.</p>
<p>Find the set of four distinct digits, <i>a</i> <img src='images/symbol_lt.gif' width='10' height='10' alt='&lt;' border='0' style='vertical-align:middle;' /> <i>b</i> &lt <i>c</i> <img src='images/symbol_lt.gif' width='10' height='10' alt='&lt;' border='0' style='vertical-align:middle;' /> <i>d</i>, for which the longest set of consecutive positive integers, 1 to <i>n</i>, can be obtained, giving your answer as a string: <i>abcd</i>.</p>

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